Measurement schemes for quantum linear equation solvers

Abstract

Solving computational fluid dynamics (CFD) problems requires the inversion of a linear system of equations, which can be done using a quantum algorithm for matrix inversion (Gilyén et al 2019 Proc. 51st Annual ACM SIGACT Symp. on Theory of Computing 193–204). However, the number of shots required to measure the output of the system can be prohibitive and remove any advantage obtained by quantum computing. In this work we propose a scheme for measuring the output of a quantum singular value transform (QSVT) matrix inversion algorithm specifically for the CFD use case. We use a quantum signal processing based amplitude estimation algorithm (Rall and Fuller 2023 Quantum7 937) and show how it can be combined with the QSVT matrix inversion algorithm. We perform a detailed resource estimation of the amount of computational resources required for a single iteration of amplitude estimation, and compare the costs of amplitude estimation with the cost of not doing amplitude estimation and measuring the whole wavefunction. We also propose a measurement scheme to reduce the number of amplitudes measured in the CFD example by focussing on large amplitudes only. We simulate the whole CFD loop, finding that thus measuring only a small number of the total amplitudes in the output vector still results in an acceptable level of overall error.